6.15.45 problem 41

Internal problem ID [2043]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS II. Exercises 7.6. Page 374
Problem number : 41
Date solved : Tuesday, September 30, 2025 at 05:22:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-9 x^{2}+5\right ) y^{\prime }+\left (-3 x^{2}+4\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*(-2*x^2+1)*diff(diff(y(x),x),x)+x*(-9*x^2+5)*diff(y(x),x)+(-3*x^2+4)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {3}{4} x^{2}-\frac {9}{64} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x^{2}-\frac {21}{128} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x^{2}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

ode=x^2*(1-2*x^2)*D[y[x],{x,2}]+x*(5-9*x^2)*D[y[x],x]+(4-3*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {9 x^4}{64}-\frac {3 x^2}{4}+1\right )}{x^2}+c_2 \left (\frac {\frac {x^2}{2}-\frac {21 x^4}{128}}{x^2}+\frac {\left (-\frac {9 x^4}{64}-\frac {3 x^2}{4}+1\right ) \log (x)}{x^2}\right ) \]

✓ Sympy. Time used: 0.529 (sec). Leaf size: 10

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(1 - 2*x**2)*Derivative(y(x), (x, 2)) + x*(5 - 9*x**2)*Derivative(y(x), x) + (4 - 3*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x^{2}} + O\left (x^{6}\right ) \]